On Weak Solutions of SDEs with Singular Time-Dependent Drift and Driven by Stable Processes

Abstract

Let $d\ge 2$. In this paper, we study weak solutions for the following type of stochastic differential equation \[ dX_{t}=dS_{t}+b(s+t, X_{t})dt, \quad X_{0}=x, \] where $(s,x)\in {\mathbb{R}}_{+}\times {\mathbb{R}}^d$ is the initial starting point, $b:{\mathbb{R}}_{+}\times {\mathbb{R}}^d\to {\mathbb{R}}^d$ is measurable, and $S=(S_t{)}_{t\ge 0}$ is a $d$-dimensional $\alpha$-stable process with index $\alpha \in (1,2)$. We show that if the $\alpha$-stable process $S$ is non-degenerate and $b\in L_{loc}^{\infty }({\mathbb{R}}_{+};L^{\infty }({\mathbb{R}}^d))+L_{loc}^q({\mathbb{R}}_{+};L^p({\mathbb{R}}^d))$ for some $p,q>0$ with $d/p+\alpha /q<\alpha -1$, then the above SDE has a unique weak solution for every starting point $(s,x)\in {\mathbb{R}}_{+}\times {\mathbb{R}}^d$.

37 pages

References in corpus (3)

• Well-posedness of the transport equation by stochastic perturbation
• Stochastic flows for Lévy processes with Hölder drifts
• Uniqueness of Stable Processes with Drift